
the bivariate Poisson distribution [1]}, so \(X_t\)  is the familiar “Poisson \(\mathrm {AR}(1)\) 
Process” of
McKenzie [2]}, [3]} (with autocorrelation \(\rho ={r_{1}}\) )
considered in Sec. (REF ). Its connection with Markov branching
processes was recognized earlier [4]}.
By Eqn. (REF ) the conditional distribution of \(X_{t+1}\) , given \({\mathcal {F}}_t:=\sigma \left\lbrace  X_s:~s\le t\right\rbrace \) , is that of the sum of \(X_t\)  independent
Bernoulli random variables with pgf \(p(s)\)  and a Poisson innovation term with
pgf \(\exp \lbrace P({r_{0}} s)-P({r_{0}})\rbrace \) ,
so the Markov process \(X_t\)  may be written recursively starting at any \(t_0\)  as
\(X_{t_0} &\sim \textsf {Po}({\theta _{+}})\\X_{t} &= \xi _t+\zeta _t,\text{\quad where }\xi _t \sim \textsf {Bi}(X_{t-1}, {r_{1}}) \text{ and }\zeta _t \sim \textsf {Po}({\theta _{t}}{r_{0}})\\\) 
